Exhaustive search, backtracking, object-oriented Queens After today, you should be able to… In SVN: …explain the concept of backtracking …solve the n-queens problem using backtracking Qu Queens ens
A taste of artificial intelligence Check out Queens from SVN
Given: a (large) set of possible solutions to a problem The “search space” Goal: Find all solutions (or an optimal solution) from that set Questions we ask: ◦ How do we represent the possible solutions? ◦ How do we organize the search? ◦ Can we avoid checking some obvious non- solutions?
Examples: Doublets, solving a maze, the “15” puzzle. Taken from: ◦ http://www.cis.upenn.edu/~matuszek/cit594- 2004/Lectures/38-backtracking.ppt
dead end ? dead end dead end ? start ? ? dead end http://www.cis.upenn.ed dead end u/~matuszek/cit594- ? 2004/Lectures/38- backtracking.ppt success!
◦ In how many ways can N chess queens be placed on an NxN grid, so that none of the queens can attack any other queen? ◦ I.e. there are not two queens on the same row, same column, or same diagonal. There is no "formula" for generating a solution. The famous computer scientist Niklaus Wirth described his approach to the problem in 1971: Progr gram am Develop opme ment nt by Stepwis ise Refine inement ment http://sunnyday.mit.edu/16.355/wirth- refinement.html#3 http://en.wikipedia.org/wiki/Queen_(chess)
In how many ways can N chess queens be placed on an NxN grid, so that none of the queens can attack any other queen? ◦ I.e. no two queens on the same row, same column, or same diagonal. Two minutes No Peeking!
1 Very naive approach. ch. Perhaps s stupid id is a b better er word! There are N queens, N 2 squares. For each queen, try every possible square, allowing the possibility of multiple queens in the same square. ◦ Represent each potential solution as an N-item array of pairs of integers (a row and a column for each queen). ◦ Generate all such arrays (you should be able to write code that would do this) and check to see which ones are solutions. ◦ Number of possibilities to try in the NxN case: ◦ Specific number for N=8: 281,474,976,710,656 281,474,976,710,656
Sligh ght improvem ovemen ent. t. There are N queens, N 2 squares. For each queen, try every possible square, notice that we can't have multiple queens on the same square. ◦ Represent each potential solution as an N-item array of pairs of integers (a row and a column for each queen). ◦ Generate all such arrays and check to see which ones are solutions. ◦ Number of possibilities to try in NxN case: ◦ Specific number for N=8: 178,462,987,637,760 (vs. . 281,474,976,710,656)
Slight ghtly ly better approach. ach. There are N queens, N columns. If two queens are in the same column, they will attack each other. Thus there must be exactly one queen per column. Represent a potential solution as an N-item array of integers. ◦ Each array position represents the queen in one column. ◦ The number stored in an array position represents the row of that column's queen. ◦ Show array for 4x4 solution. Generate all such arrays and check to see which ones are solutions. Number of possibilities to try in NxN case: Specific number for N=8: 16,777,216
Still better ter approac roach There must also be exactly one queen per row. Represent the data just as before, but notice that the data in the array is a set! ◦ Generate each of these and check to see which ones are solutions. ◦ How w to generate? erate? A good thing to think about. ◦ Number of possibilities to try in NxN case: ◦ Specific number for N=8: 40,320
Ba Backtrackin ktracking g soluti ution on Instead of generating all permutations of N queens and checking to see if each is a solution, we generate "partial placements" by placing one queen at a time on the board Once we have successfully placed k<N queens, we try to extend the partial solution by placing a queen in the next column. When we extend to N queens, we have a solution.
Play the game: ◦ http://homepage.tinet.ie/~pdpals/8queens.htm See the solutions: ◦ http://www.dcs.ed.ac.uk/home/mlj/demos/queens
>java RealQueen 5 SOLUTION: 1 3 5 2 4 SOLUTION: 1 4 2 5 3 SOLUTION: 2 4 1 3 5 SOLUTION: 2 5 3 1 4 SOLUTION: 3 1 4 2 5 SOLUTION: 3 5 2 4 1 SOLUTION: 4 1 3 5 2 SOLUTION: 4 2 5 3 1 SOLUTION: 5 2 4 1 3 SOLUTION: 5 3 1 4 2
2-5 Board configuration represented by a linked list of Queen objects Fields of RealQue alQueen: column row neighbor Designed by Timothy Budd http://web.engr.oregonstate.edu/~budd/Books/oopintro3e/info/slides/chap06/java.htm
Each queen sends messages directly to its immediate neighbor to the left (and recursively to all of its left neighbors) Return value provides information concerning all of the left neighbors: Example: neighbor.canAttack(currentRow, col) ◦ Message goes to the immediate neighbor, but the real question to be answered by this call is ◦ "Hey, neighbors, can any of you attack me if I place myself on this square of the board?"
6-10 10 findFirst() findNext() canAttack(int row, int col) Your job (part of WA6): Underst erstand nd the job of each of these se method hods. s. Javadoc adoc from m the Qu Queen n interface rface ca can help Fill in the (recursive) details in the RealQueen class Debug ug More detail ils s on next t slide de
1. Queen asks its neighbors to find the first position in which none of them attack each other ◦ Found? Then queen tries to position itself so that it cannot be attacked. 2. If the rightmost queen is successful, then a solution has been found! The queens cooperate in recording it. 3. Otherwise, the queen asks its neighbors to find the next position in which they do not attack each other 4. When the queens get to the point where there is no next non-attacking position, all solutions have been found and the algorithm terminates And recu cursi rsion on does its magic! ic!
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